91Ó°ÊÓ

Find the area enclosed by the given curves. $$ y=x^{2}+1, y=x, x=-1, x=1 $$

Evaluate the definite integrals. \(\int_{0}^{1}(2 x-1)^{9} d x\)

In Exercises 15 through 18 , find the consumers' surplus, using the given demand equations and the equilibrium price \(p_{0}\). $$ D(x)=20-x^{2}, p_{0}=4 $$

In Exercises 11 through 16 we will consider approximations to the distance traveled by an object with velocity \(v=f(t)\) on the given interval \([a, b] .\) The answers are most easily obtained by using the Integration Kit on the disk provided. For each of these exercises, do the following: (a) For \(n=4,\) make a sketch that illustrates the left-and right hand sums, showing clearly the four rectangles and \(x_{0}, x_{1}\), \(x_{2}, x_{3},\) and \(x_{4}\) (b) For \(n=4,\) find the left-and right-hand sums. Also calculate the difference between the upper and lower estimates. Calculate the average of the two sums. (c) Repeat parts (a) and (b) with \(n=8\) and \(40 .\) Calculate the average of the two sums. Compare your answers with those in part (b). $$ v=f(t)=t^{3}-6 t^{2}+9 t-1,[0,4] $$

Estimate the given definite integrals by finding the left- and right-hand sums for \(n=80,160,320\), and \(640 .\) These exercises are most easily done by using the interactive illustration found in the Integration Kit. $$ v=f(t)=\frac{2}{1+e^{-t}},[0,5] $$

Estimate the given definite integrals by finding the left- and right-hand sums for \(n=80,160,320\), and \(640 .\) These exercises are most easily done by using the interactive illustration found in the Integration Kit. $$ v=f(t)=0.001 t^{3}-0.03 t^{2}+0.3 t+1,[0,25] $$

Find the area enclosed by the given curves. $$ y=x+2, y=\sqrt[3]{x}, x=-1, x=1 $$

In Exercises 1 through 38 , find the antiderivatives. $$ \int\left(3 u^{2}+4 u^{3}\right) d u $$

In Exercises 11 through 16 we will consider approximations to the distance traveled by an object with velocity \(v=f(t)\) on the given interval \([a, b] .\) The answers are most easily obtained by using the Integration Kit on the disk provided. For each of these exercises, do the following: (a) For \(n=4,\) make a sketch that illustrates the left-and right hand sums, showing clearly the four rectangles and \(x_{0}, x_{1}\), \(x_{2}, x_{3},\) and \(x_{4}\) (b) For \(n=4,\) find the left-and right-hand sums. Also calculate the difference between the upper and lower estimates. Calculate the average of the two sums. (c) Repeat parts (a) and (b) with \(n=8\) and \(40 .\) Calculate the average of the two sums. Compare your answers with those in part (b). $$ v=f(t)=t \sin t,[0,2 \pi] $$

Find the indefinite integral. $$ \int \frac{\ln 2 x}{x} d x $$